Information Acquisition and Response in Peer-Effects Networks

Transcription

1 Information Acquisition and Response in Peer-Effects Networks C. Matthew Leister Monash University Conference on Economic Networks and Finance LSE, December 11, 2015

2 Individuals/firms face heterogeneous incentives to acquire and respond to information.

3 Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs

4 Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs Strategic position

5 Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs Strategic position Dual role of information:

6 Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs Strategic position Dual role of information: 1. infer the state of the world,

7 Individuals/firms face heterogeneous incentives to acquire and respond to information. Idiosyncratic values/costs Strategic position Dual role of information: 1. infer the state of the world, 2. in equilibrium, infer the observations and subsequent actions of neighbors.

8 Peer-effects networks with incomplete information u i (x 1,...,x N ) = a i +ω + σ ik x k x i 1 2 σ iixi 2 k i }{{}}{{} O.C. to x i marginal value to x i

9 u i (x 1,...,x N ) = (a i +ω + k i σ ikx k ) x i 1 2 σ iix 2 i A competitive supply chain up stream firm + + down stream firm vert. integ. firm ω : demand for novel product x firm : production

10 u i (x 1,...,x N ) = (a i +ω + k i σ ikx k ) x i 1 2 σ iix 2 i Traders with heterogeneous funding constraints deep pockets + liquidity const. ω : long term asset value x trader : market order/inventory

11 Basic questions (1) How does heterogeneity in strategic positioning influence the incentives to acquire information?

12 Basic questions (1) How does heterogeneity in strategic positioning influence the incentives to acquire information? (2) Who over and who under acquires information? Who gains to influence others beliefs?

13 Positive results Information response game EQ value to information. Equilibrium properties: a. game on correlation-adjusted network (second stage), b. negative responses (second stage), c. multiple information acquisition equilibria (first stage).

14 Welfare results 1. Extent of symmetry among pair-wise peer effects drives direction of two inefficiencies: a. informational externalities (network charact.: in-walks), b. strategic value to information acquisition (network charact.: closed-walks). 2. Symmetric networks (for e.g.) a. bunching for moderate peer effects: equilibrium information asymmetries inefficiently low, b. significant strategic substitutes: acquisition of negative responders inefficiently low, c. positive strategic distortion connectedness in network. 3. Antisymmetric networks: inefficiencies reverse.

15 Policy implications Transparency-based policy: targeted certification of information investments.

16 Literature Network games with incomplete information: Calvó-Armengol & de Martí (2007,2009), Calvó-Armengol, de Martí, Prat (2015), de Martí & Zenou (2015). Coordination games with endogenous information: Novshek & Sonnenschein (1983,1988), Vives (1988,2008), Hauk & Hurkens (2007). Morris & Shin (2002), Hellwig & Veldkamp (2009), Myatt & Wallace (2012,2013), Colombo, Femminis, & Pavan (2014). Finance: Grossman & Stiglitz (1980), Kyle (1985,1989), Babus & Kondor (2013).

17 Timeline of the game each i chooses information quality e i [0,1] at cost κ i (e i ) each i observes signal θ i, then chooses action x i R state ω Ω observed, each i s u i (x ω) realized t = 1 t = 2 t = 2+

18 Model primitives: second stage (t = 2) Each i chooses x i R, yielding i s payoffs (t = 2): u i (x ω) = ω +ρ σ ik x k x i 1 2 x2 i, k i where ω Ω R, σ ij R for each i,j, and ρ [0,1],

19 Model primitives: second stage (t = 2) Each i chooses x i R, yielding i s payoffs (t = 2): u i (x ω) = ω +ρ σ ik x k x i 1 2 x2 i, k i where ω Ω R, σ ij R for each i,j, and ρ [0,1], i observes signal θ i Θ R of quality e i [0,1],

20 Model primitives: second stage (t = 2) Each i chooses x i R, yielding i s payoffs (t = 2): u i (x ω) = ω +ρ σ ik x k x i 1 2 x2 i, k i where ω Ω R, σ ij R for each i,j, and ρ [0,1], i observes signal θ i Θ R of quality e i [0,1], Pure strategy: X i : Θ [0,1] R.

21 Model primitives: second stage (t = 2) Each i chooses x i R, yielding i s payoffs (t = 2): u i (x ω) = ω +ρ σ ik x k x i 1 2 x2 i, k i where ω Ω R, σ ij R for each i,j, and ρ [0,1], i observes signal θ i Θ R of quality e i [0,1], Pure strategy: X i : Θ [0,1] R. Assumption 1 (I [s ij σ ij ]) 1 is well defined for every s [0,1] N(N 1).

22 Model primitives: first stage (t = 1) Each i = 1,...,N privately invests in information quality e i [0,1]. i s cost of information quality κ i ( ) C 2 satisfies κ i (0),κ i (0) = 0, with non-decreasing κ i (e i) 0. Assumption 2 For v 0 > 0, there exists an unique e i (0,1) solving v 0 e i = κ i (e i ).

23 Model primitives: first stage (t = 1) Each i = 1,...,N privately invests in information quality e i [0,1]. i s cost of information quality κ i ( ) C 2 satisfies κ i (0),κ i (0) = 0, with non-decreasing κ i (e i) 0. Assumption 2 For v 0 > 0, there exists an unique e i (0,1) solving v 0 e i = κ i (e i ). All conditions satisfied for normal state and signals case.

24 Model primitives: beliefs and expectations Belief: µ i (e i ), density function over e i [0,1] N 1.

25 Model primitives: beliefs and expectations Belief: µ i (e i ), density function over e i [0,1] N 1. Consistency: µ i (e i ) = 1 for t = 1 for given e i, with µ i (e i ) = 0 otherwise.

26 Model primitives: beliefs and expectations Belief: µ i (e i ), density function over e i [0,1] N 1. Consistency: µ i (e i ) = 1 for t = 1 for given e i, with µ i (e i ) = 0 otherwise. E1. E i [ω] = E i [θ i ] = 0, v 0 := E i [ ω 2 ] = E i [ θ 2 i e i ], E2. E3. E i [ω θ i,e i ] = e i θ i, E i [θ j θ i,e i,e j ] = e i e j θ i, for each e i [0,1].

27 Equilibrium facts Theorems 1. Multiple IAE e may exist even with a unique IRE β for each e.

28 Equilibrium facts Theorems 1. Multiple IAE e may exist even with a unique IRE β for each e. 2. Significant strategic substitutes: can have β i < 0.

29 Equilibrium facts Theorems 1. Multiple IAE e may exist even with a unique IRE β for each e. 2. Significant strategic substitutes: can have β i < Significant peer effects required for 1. or 2. to obtain.

30 Equilibrium facts Theorems 1. Multiple IAE e may exist even with a unique IRE β for each e. 2. Significant strategic substitutes: can have β i < Significant peer effects required for 1. or 2. to obtain. Proposition Under Assumptions 1 and 2, there exists a ρ > 0 such that for ρ [0, ρ), a unique IAE e with β i > 0 for all i obtains.

31 Welfare

32 Welfare For any e, giving X : ν i (X e) := E i [u i (X θ i,e i,µ i) e i,µ i] κ i (e i ). = 1 2 v 0β 2 i κ i (e i ).

33 Welfare: marginal inefficiencies Define the utilitarian problem: max ν k (X e). e [0,1] N k

34 Welfare: marginal inefficiencies Define the utilitarian problem: max ν k (X e). e [0,1] N k e i k ν k (X e) = ν i (X e) e i + β k,k i k i ν i (X e) βk βk + e i k i ν k (X e) βk βk. e i

35 Welfare: marginal inefficiencies Define the utilitarian problem: max ν k (X e). e [0,1] N k e i k ν k (X e) β = (v 2 ) i 0 κ (e i ) + v 0 β i e i ρσ ik e k βk + v 0 e i e i }{{} k i k i = 0 in IAE e f.o.c. }{{} = 0 in public acquisition eq. e pb f.o.c. β k βk e. i } {{ } = 0 in planner s solution e pl f.o.c.

36 Welfare: marginal inefficiencies Define the utilitarian problem: max ν k (X e). e [0,1] N k e i k ν k (X e) β = (v 2 ) i 0 κ (e i ) + v 0 β i e i ρσ ik e k βk + v 0 β k βk e i e i e. i k i k i }{{}}{{} (marginal) strategic value (marginal) externalities

37 Welfare: marginal inefficiencies Define the utilitarian problem: max ν k (X e). e [0,1] N k e i k ν k (X e) β = (v 2 ) i 0 κ (e i ) + v 0 β i e i ρσ ik e k βk + v 0 β k βk e i e i e. i k i k i }{{}}{{} ξi st (X e) ξi ex (X e)

38 Welfare: marginal inefficiencies Define the utilitarian problem: max ν k (X e). e [0,1] N k e i k ν k (X e) β = (v 2 ) i 0 κ (e i ) + v 0 β i e i ρσ ik e k βk e i e + v 0 β k βk i e. i k i k i }{{} (marginal) public-value

39 Welfare: marginal inefficiencies Theorem (marginal inefficiencies) For information qualities e, consistent beliefs µ and IRE X : i (e,x βi 2 ) = 2v 0 1 ii e ΣI e (I I e ΣI e ) 1 I e ΣI e 1 i, ξ st e i ξ ex i (e,x ) = 2v 0 β i e i (β β i 1 i ) I e ΣI e (I I e ΣI e ) 1 1 i.

40 Welfare: marginal inefficiencies Theorem (marginal inefficiencies) For information qualities e, consistent beliefs µ and IRE X : i (e,x βi 2 ) = 2v 0 1 ii e ΣI e (I I e ΣI e ) 1 I e ΣI e 1 i, ξ st e i ξ ex i (e,x ) = 2v 0 β i e i (β β i 1 i ) I e ΣI e (I I e ΣI e ) 1 1 i. ξ st i (e,x ) 1 i ( ([e i e j ρσ ij ] i j ) )1 τ i : τ=2 summation of closed walks on [e i e j ρσ ij ] i j beginning and ending on i.

41 Welfare: marginal inefficiencies Theorem (marginal inefficiencies) For information qualities e, consistent beliefs µ and IRE X : i (e,x βi 2 ) = 2v 0 1 ii e ΣI e (I I e ΣI e ) 1 I e ΣI e 1 i, ξ st e i ξ ex i (e,x ) = 2v 0 β i e i (β β i 1 i ) I e ΣI e (I I e ΣI e ) 1 1 i. ξ ex ( i (e,x ) (β βi 1 i ) ([e i e j ρσ ij ] i j ) )1 τ i : τ=1 summation of walks on [e i e j ρσ ij ] i j beginning with j and ending on i, weighted by β j and aggregate over j i.

42 Example: three-player symmetric network, common κ.80 e i e e pl +e pb i i ei i

43 Example: three-player symmetric network, common κ e i / e / e pl +e pb i i ei i e pl 1 = 0

44 Example: three-player symmetric network, common κ e i / e 1/ / e pl +e pb i i ei i e pl 1 = epb 1 = e 1 = 0

45 Example: two-player antisymmetric network, common κ.75 e i + e 1/ / e pl +e pb i i ei i

46 Welfare and policy design

47 Welfare and the neutral player symmetric networks ξ pl i (II) (III) e i (I) β i antisymmetric networks ξ pl i (VI) (IV) (V) e i β i

48 Market efficiency in liquidity crises conclusion

49 Market efficiency in liquidity crises N = 8 traders comprise non-trivial share of market.

50 Market efficiency in liquidity crises N = 8 traders comprise non-trivial share of market. x i : i s inventory/market order (e.g. Kyle (1985)); x := 8 i=1 x i.

51 Market efficiency in liquidity crises N = 8 traders comprise non-trivial share of market. x i : i s inventory/market order (e.g. Kyle (1985)); x := 8 i=1 x i. t = 2 market price φ( x) = A+B x, B > 0.

52 Market efficiency in liquidity crises N = 8 traders comprise non-trivial share of market. x i : i s inventory/market order (e.g. Kyle (1985)); x := 8 i=1 x i. t = 2 market price φ( x) = A+B x, B > 0. ω: risky asset s long term value.

53 Market efficiency in liquidity crises N = 8 traders comprise non-trivial share of market. x i : i s inventory/market order (e.g. Kyle (1985)); x := 8 i=1 x i. t = 2 market price φ( x) = A+B x, B > 0. ω: risky asset s long term value. t = 2 payoffs: u i (x ω) = (ω +p i φ( x))x i xi 2 = ω +p i A+p i B k i x k x i (1 p i B)x 2 i.

54 Market efficiency in liquidity crises Liquidity flush market: p i < 0 for each unconstrained i.

55 Market efficiency in liquidity crises Liquidity flush market: p i < 0 for each unconstrained i. Market crowding in information acquisition.

56 Market efficiency in liquidity crises Liquidity flush market: p i < 0 for each unconstrained i. Market crowding in information acquisition. Traders set e i,β i < e (region (II)): over-acquire; over exertion in informationally inefficient markets.

57 Market efficiency in liquidity crises Liquidity crises: Liquidity spirals à la Brunnermeier and Pedersen (2009) upward sloping demand.

58 Market efficiency in liquidity crises Liquidity crises: Liquidity spirals à la Brunnermeier and Pedersen (2009) upward sloping demand. p i > 0 for liquidity-constrained trader i.

59 Market efficiency in liquidity crises Market structure: p i B > 0 p i B < constrained traders unconstrained traders

60 Market efficiency in liquidity crises Liquidity crisis paradigm shift: Constrained traders set e i,β i > e 1. Flush market: antisymmetric relationships over-acquire. 2. Crisis: symmetric relationships under-acquire.

61 Market efficiency in liquidity crises e i antisymmetric peer effects ecnst. pl symmetric peer effects e cnst e # cnst.

62 Market efficiency in liquidity crises Liquidity crisis paradigm shift: Constrained traders set ei,β i > e 1. Flush market: antisymmetric relationships over-acquire. 2. Crisis: symmetric relationships under-acquire. Unconstrained traders set ei,β i < e 1. Flush market: symmetric relationships over-acquire. 2. Crisis: antisymmetric relationships under-acquire. 3. Extreme crisis: few unconstrained traders set ei,β i < 0.

63 Market efficiency in liquidity crises e i symmetric peer effects e unc. antisymmetric peer effects e pl unc. e negative response # cnst.

64 Policy suggestion in liquidity crises Constrained traders impose symmetric, positive informational externalities on each other: under acquire, with positive strategic values...

65 Policy suggestion in liquidity crises Constrained traders impose symmetric, positive informational externalities on each other: under acquire, with positive strategic values... Couple stress-tests with certification of information investments of constrained traders.

66 Conclusions 1. Introduce problem of costly information acquisition into new context: general network of peer effects. 2. Symmetric networks: a. Equilibrium information inefficiently symmetric. b. Players moving against their information do so too little. c. Strategic values to information are positive. 3. Direction of welfare and strategic motives determined by network position and extent of symmetry in relationships: direction of inefficiencies reverse in antisymmetric networks. 4. Information externalities and position : β i w.r.t. e i and origin, Strategic values and position : connectedness.

67 Conclusions II 1. Liquidity crisis paradigm shift: over acquisition of information in liquid markets, under acquisition in constrained markets. 2. Unconstrained shorters in crisis: inefficient. 3. Transparency-based policy intervention: stress test with information investment certification.

68 Equilibrium characterization

69 Equilibrium characterization Theorem (t = 2 information-response equilibrium (IRE)) Under Assumption 1, for any e and consistent µ there exists a unique linear IRE of the form: X = [X i (θ i e i )] = [β i θ i ], where each β i solves β i = e i + k i e ie k ρσ ik β k : β := (I [e i e j ρσ ij ] i j ) 1 e = ([e i e j ρσ ij ] i j ) τ e. τ=0

70 Equilibrium characterization Theorem (t = 2 information-response equilibrium (IRE)) Under Assumption 1, for any e and consistent µ there exists a unique linear IRE of the form: X = [X i (θ i e i )] = [β i θ i ], where each β i solves β i = e i + k i e ie k ρσ ik β k : β i β := (I [e i e j ρσ ij ] i j ) 1 e = ([e i e j ρσ ij ] i j ) τ e. τ=0 : i s informational centrality (weighted Bonacich centrality).

71 Equilibrium characterization Theorem (t = 1 information-acquisition equilibrium (IAE)) Under Assumption 1, for IRE X and consistent beliefs µ there exists a (generically unique ) IAE e. For any such IAE, and i with ei (0,1): βi 2 v 0 ei = κ i (ei ). Back

72 Equilibrium characterization Theorem (t = 1 information-acquisition equilibrium (IAE)) Under Assumption 1, for IRE X and consistent beliefs µ there exists a (generically unique ) IAE e. For any such IAE, and i with ei (0,1): βi 2 v 0 ei = κ i (ei ). Back e i i